Theorem exiftru 1891

Description: Rule of existential generalization, similar to universal generalization ax-gen 1722, but valid only if an individual exists. Its proof requires ax-6 1888 but the equality predicate does not occur in its statement. Some fundamental theorems of predicate logic can be proven from ax-gen 1722, ax-4 1737 and this theorem alone, not requiring ax-7 1935 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.)

Hypothesis

Ref	Expression
exiftru.1	|- ph

Assertion

Ref	Expression
exiftru	|- E. x ph

Proof of Theorem exiftru

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax6ev 1890	. 2 |- E. x x = y
2		exiftru.1	. . 3 |- ph
3	2	a1i 11	. 2 |- ( x = y -> ph )
4	1, 3	eximii 1764	1 |- E. x ph

Syntax hints:   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  19.2  1892  bj-extru  32654  ac6s6  33980